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 IRMA Lectures in Mathematics and Theoretical Physics 2007; 802 pp; hardcover Volume: 11 ISBN-10: 3-03719-029-9 ISBN-13: 978-3-03719-029-6 List Price: US$128 Member Price: US$102.40 Order Code: EMSILMTP/11 See also: Families of Riemann Surfaces and Weil-Petersson Geometry - Scott A Wolpert Handbook of Teichmüller Theory: Volume II - Athanase Papadopoulos Handbook of Teichmüller Theory: Volume III - Athanase Papadopoulos Handbook of Teichmüller Theory: Volume IV - Athanase Papadopoulos The Teichmüller space of a surface was introduced by O. Teichmüller in the 1930s. It is a basic tool in the study of Riemann's moduli spaces and the mapping class groups. These objects are fundamental in several fields of mathematics, including algebraic geometry, number theory, topology, geometry, and dynamics. The original setting of Teichmüller theory is complex analysis. The work of Thurston in the 1970s brought techniques of hyperbolic geometry to the study of Teichmüller space and its asymptotic geometry. Teichmüller spaces are also studied from the point of view of the representation theory of the fundamental group of the surface in a Lie group $$G$$, most notably $$G=\mathrm{PSL}(2,\mathbb{R})$$ and $$G=\mathrm{PSL}(2,\mathbb{C})$$. In the 1980s, there evolved an essentially combinatorial treatment of the Teichmüller and moduli spaces involving techniques and ideas from high-energy physics, namely from string theory. The current research interests include the quantization of Teichmüller space, the Weil-Petersson symplectic and Poisson geometry of this space as well as gauge-theoretic extensions of these structures. The quantization theories can lead to new invariants of hyperbolic 3-manifolds. The purpose of this handbook is to give a panorama of some of the most important aspects of Teichmüller theory. The handbook should be useful to specialists in the field, to graduate students, and more generally to mathematicians who want to learn about the subject. All the chapters are self-contained and have a pedagogical character. They are written by leading experts in the subject. A publication of the European Mathematical Society. Distributed within the Americas by the American Mathematical Society. Readership Graduate students and research mathematicians interested in analysis. Table of Contents A. Papadopoulos -- Introduction to Teichmüller theory, old and new Part A. The metric and the analytic theory, 1 G. D. Daskalopoulos and R. A. Wentworth -- Harmonic maps and Teichmüller theory A. Papadopoulos and G. Théret -- On Teichmüller's metric and Thurston's asymmetric metric on Teichmüller space R. C. Penner -- Surfaces, circles, and solenoids J.-P. Otal -- About the embedding of Teichmüller space in the space of geodesic Hölder distributions W. J. Harvey -- Teichmüller spaces, triangle groups and Grothendieck dessins F. Herrlich and G. Schmithüsen -- On the boundary of Teichmüller disks in Teichmüller and in Schottky space Part B. The group theory, 1 S. Morita -- Introduction to mapping class groups of surfaces and related groups L. Mosher -- Geometric survey of subgroups of mapping class groups A. Marden -- Deformations of Kleinian groups U. Hamenstädt -- Geometry of the complex of curves and of Teichmüller space Part C. Surfaces with singularities and discrete Riemann surfaces C. Charitos and I. Papadoperakis -- Parameters for generalized Teichmüller spaces M. Troyanov -- On the moduli space of singular euclidean surfaces C. Mercat -- Discrete Riemann surfaces Part D. The quantum theory, 1 L. O. Chekhov and R. C. Penner -- On quantizing Teichmüller and Thurston theories V. V. Fock and A. B. Goncharov -- Dual Teichmüller and lamination spaces J. Teschner -- An analog of a modular functor from quantized Teichmüller theory R. M. Kashaev -- On quantum moduli space of flat $$\mathrm{PSL}_2(\mathbb{R})$$-connections List of contributors Index